Optimal. Leaf size=56 \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac {1}{n},-b \log (F);1+\frac {1}{n};-\frac {d x^n}{c}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2274, 12, 246, 245} \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac {1}{n},-b \log (F);1+\frac {1}{n};-\frac {d x^n}{c}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 245
Rule 246
Rule 2274
Rubi steps
\begin {align*} \int F^{a+b \log \left (c+d x^n\right )} \, dx &=\int F^a \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=F^a \int \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=\left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)}\right ) \int \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx\\ &=F^a x \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac {1}{n},-b \log (F);1+\frac {1}{n};-\frac {d x^n}{c}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 1.48 \[ -\frac {x \left (-\frac {d x^n}{c}\right )^{-1/n} \left (c+d x^n\right ) F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (\frac {n-1}{n},b \log (F)+1;b \log (F)+2;\frac {d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b \log \left (d x^{n} + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{b \log \left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int F^{b \ln \left (d \,x^{n}+c \right )+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{b \log \left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 58, normalized size = 1.04 \[ \frac {F^a\,x\,{\left (c+d\,x^n\right )}^{b\,\ln \relax (F)}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n},-b\,\ln \relax (F);\ \frac {1}{n}+1;\ -\frac {d\,x^n}{c}\right )}{{\left (\frac {d\,x^n}{c}+1\right )}^{b\,\ln \relax (F)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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